Chapter 6

Focal and parallel surfaces

For an embedding
X: U -> R3 of a domain
U of R2 into 3-space, the parallel
surface at distance r is the image of the parallel
mapXr: U -> R3
defined by
Xr(P) = X(P) + rN(P),
where
N = (Xx x Xy)/|Xx x Xy|
is the unit normal of X. The parallel surface can be thought
of as a "wave front" at distance r from the image of X.
Let S(Xr) be the set of singular
points of Xr.

S(Xr) = {PU | rank
d(Xr)
Then PS(Xr) if and only if r is a
principal radius of curvature of X at P with focal point
Xr(P). So the union of the singular
sets of parallel surfaces of X is the focal surfaceF.

Let be the normal bundle of X:

= {(V, P) R3 x U | V is normal to X at P}

and let : -> R3 be defined by (V, P) = X(P) + V.
The focal surface F is the set of critical values of . If the normal lines are thought of as rays of light
streaming out from the surface, then the focal surface is the
"caustic" of this bundle of rays. In purely geometric terms, the
focal surface is the envelope of the family of normal lines of the
embedding X[T2].

Gauss thought of his mapping N as assigning to each point of
a surface a point on the sphere at infinity, analogous to the
celestial sphere used in navigation and surveying [Ba, p. 45]. Let
p: R3 - {0} -> S2
be radial projection. For r sufficiently large, the parallel
surface to X at distance r does not pass through the origin, so
p ° Xr is defined, and the limit
as r goes to infinity of (p ° Xr)
is N. In other words, the Gauss map is the parallel map at
infinity. Another way to see this is to consider the family of
maps
It: U -> R3,
0 t
It(P) = (1-t) X(P) + tN(P)

= (1-t)[X(P) + t/(1-t) N(P)]

The family It is the linear interpolation
between the embedding X and its Gauss map N. Each map
I for 0
t Xr,
r = t/(1 - t) (cf. [Ba]).
Thus the critical values of the Gauss map are the focal points at
infinity.

Now consider an immersion
X: Mn -> Rn+1
of the smooth n-manifold M in Euclidean
(n + 1)-space. Let be the unit
normal bundle of X:

= {(V, P) S2 x M|V is
normal to X at P}

As observed in chapter 5, is the oriented double
cover of M. For each real number r > 0, the
parallel map to X at distance r is

Xr: -> Rn+1,
Xr(V, P) = X(P) + rV

The image of Xr is the parallel hypersurface
of X at radius r. The union of the singular sets of
parallel surfaces of X is the focal hypersurface F. If
is the normal bundle of X, and : -> Rn+1, (V, P) = X(P) + V,
the focal hypersurface F is the set of critical values of .

For each point ARn+1 let
DA: M -> R be
the radial distance squared function from A:

DA = |A - X(P)|2

For almost all ARn+1, the function
DA is a nondegenerate Morse function on
M[Mi, p. 36]. Following Thom [T4] and Porteous [Por1], consider
the family of all such radial functions parametrized by
Rn+1:

D : Rn+1n -> Rn+1 x R,
D(A, P) =
(A, DA(P))

The critical set C of the family D is the set of pairs
(A, P) such that the vector
A - X(P) is normal to the immersion
X at P, so C is identified with the normal bundle
, and the catastrophe map :C -> Rn+1 is
just the map : -> Rn+1. Therefore
the focal surface ofXis the bifurcation set of the
family of radial distance functions D.

To relate the map : -> Rn+1 and the
Gauss map N: -> Sn,
we fit the two families D and together,
following Looijenga [Lo], [Wa2, p.713]. Consider the family of
functions

Theorem 6.1 (Looijenga). Let M2 be a
smooth surface. For an open dense subset B of the space of
immersions
X: M2 -> R3,
the germ at (A, P) of the family L is a
versal unfolding of the germ of LA at
P for all (A, P) S3 x M2.

Proof See [Lo] and [Wa2]. Looijenga proves the equivalent
statement that the germ of the mapping
L:S3 x M2 -> S3 x R
is stable. (the subset B consists of all immersions whose jet
extensions are transverse to a certain Whitney stratification of a jet
space).

For XB, the
singularities of the focal set F are cuspidal edges,
swallowtails, elliptic umbilics, and hyperbolic umbilics. The
umbilic singularities occur precisely at the foci of the umbilic
points of the immersion. A discussion of the geometry of the
focal set can be found in Porteous' paper [Por1], which was a starting
point for our research on the extrinsic geometry of surfaces.
Porteous calls the cuspidal edges of the focal surface ribs.
The corresponding curves on the surface M are the ridges of the
immersion X (cf. chapter 3). If the map : -> R3 has a cusp at
(V, P) , then P is a ridge point of X, and
1/|V| is the principal curvature associated with the ridge at
P.

This description does not include those ridge points with
associated principal curvature zero, which correspond to ribs at
infinity. To include these points, consider the bifurcation set
F Sn+1 of
the family L. For the diffeomorphism
F: Sn+1+ -> Rn+1,
f((a1, ..., an+1)) =
1/an+2 (a1, ...,
an+1), we have f(F~S3+) = F. For XB, the singularities of
F~ are cuspidal edges, swallowtails, elliptic
umbilics, and hyperbolic umbilics. If the catastrophe map ~ of L has a cusp at
(A, P) Sn+1 x Mn, then
P is a ridge point of X with associated principal
curvature an+2/|(a1, ...,
an+1)|. The singularities of F~Sn+10
correspond to singularities at infinity of F.

For example, the focal surface of the monkey saddle has an elliptic
umbilic at infinity. (the monkey saddle is an example of an immersion
X: M2 -> R2
such that X is in B but X is not in A.
The inclusion of the unit sphere in R3 is an example
of an immersion in A but not B.)

The following theorem implies theorem 3.1(d).

Theorem 6.2 If P is a cusp of the Gauss map of
X: M2 -> R3,
then P is a ridge point of X with associated principal
curvature zero. If XA
then the cusps of the Gauss map of X are the only points of M
with this property.

ProofP is a cusp of the Gauss mapping of X if
and only if (V, P) is a cusp of the catastrophe map
of the family , where V
is a unit normal vector to X at P. The 2-parameter
family is equivalent to the restriction of the
3-parameter family L to
S30 x M2.
therefore if has a cusp at
(V, P), then ~ has a
cusp at (i(V), P), where
i: S2 -> S3 is the
inclusion and is the catastrophe map of the
family L. So P is a ridge point of X with
associated principal curvature zero. Moreover, the ridge curve
crosses the parabolic curve transversely at P.

If XA then the germ
at (V, P) of is either
regular, a fold, or a cusp. If one of the principal curvatures of
X at P is zero, then the germ at
(V, P) of is a fold or a cusp.
If it is a fold, then the germ of ~ at
(i(V), P) is a fold, so P is not a ridge
point.

Theorem 6.3 If P is a cusp of the Gauss map of
X: M2 -> R3,
then given > 0 and
d > 0, there exists a point QU and a D > d
such that |P - Q| and Q is a swallowtail point of the parallel
surface to X at distance D. If XA then the cusps of the Gauss map
X are the only points of M with this property.

Proof The parallel map Xr has a
cusp at (V, Q) if and only if the map : -> R3 has a fold at
(rV, Q). So a parallel surface has a
cuspidal edge only where it meets the focal surface, and swallowtails
only where it meets the cuspidal edge of the focal surface.

If P is a Gaussian cusp, then P is a ridge point with
associated principal curvature zero (theorem 6.2). So a point
Q near P on the ridge curve through P has the
desired property. If XA
then the cusps of the Gauss map are the only points with this
property, by theorem 6.2.

The following corollary implies theorem 3.1(f).

Corollary 6.4 If P is a cusp of the Gauss map of
X: M2 -> R3,
then for any point A in R3 which is not on
the tangent plane to X at P, the point P is a
swallowtail point of the pedal surface of X from A. If
XA then the cusps of the
Gauss map X are the only points of M with this property.

Proof Consider
f: S2 x R -> R3,
f(V, t) = A + tV.
Since the Gauss map N: -> S2 is the catastrophe map
of the family : S2 x M -> S2 x R,
the cusps of N are the swallowtails of |. But f((M)) is precisely the pedal surface of X
from A.

Notice that this result gives a more visual proof of the bitangent
plane characterization of Gaussian cusps, since a swallowtail point of
the pedal surface corresponds to a limit of bitangent planes of
X.

The point P is a critical point of the distance function
DA if and only if the sphere through
X(P) centered at A is tangent to the immersion
X at P. The point P is a degenerate critical
point of DA if and only if this sphere
SA has unusual contact with X at
P. For example, if A is a focal point of X at
P then DA has Milnor number two at
P, and SA has stationary contact with
X at P. As the point A goes to infinity in the
direction VS2, the sphere SA
approaches the plane perpendicular to V, and spherical contact
becomes the planar contact discussed at the end of chapter 5.

Lagrangian singularities are generalizations of the singularities
which occur in the normal map : N -> Rn+1
of an immersion
X: Mn -> Rn+1.
The map is an example of a Lagrange mapping,
and the set of critical values of (the envelope of
the family of normal lines of X, i.e. the focal surface of
X) is its caustic[A3].

Recall that the normal map is the catastrophe
map of the family D of radial distance-squared functions. In
general, if
F: Q x M -> Q x R
is a family of real-valued functions on M parametrized by
Q, then the catastrophe map : C -> Q is Lagrangian
(cf. [Wa2, lemma 1, p. 716]). The caustic of is
the bifurcation set of the family F. In particular, the Gauss
mapping of an immersed hypersurface of Euclidean space is a Lagrange
mapping, since it is the catastrophe map of the family of projections
to lines (Chpater 5). (Cf. [Wa2, prop. 4, p. 720])

Legendre singularities are generalizations of the singularities
which occur in parallel surfaces of an immersed hypersurface of
Euclidean space. The parallel map
Xr: M -> Rn+1
of the immersion
X: M -> Rn+1 is
an example of a Legendre mapping, and its image (the parallel
hypersurface of X at distance r) is its front[A2].

If
F: Q x M -> Q x R
is a family of functions with critical set CQ x M, the map
F|C is Legendre [A2, thm. 18, p. 33]. For
example, for an immersion
X: Mn -> Rn+1,
consider the family
F:Sn x M -> Sn x R
of projections to lines. Choose a point ARn+1 and define
f: Sn x R -> Rn+1
by
f(V, t) = A +otV.
The critical set C of F is M, and
f(F(C)) is the pedal surface (from the point
A) of the immersion X. Thus the pedal surface of an
immersion has Legendre singularities (away from the point A).

A survey of the classification of simple Lagrange and Legendre
singularities is given by Arnold in his paper "Critical points of
smooth functions" [A3]. An excellent description of the geometry and
physics of Lagrange and Legendre singularities is given in Arnold's
book Mathematical Methods of Classical Mechanics[A4].

Arnold has communicated to us the following elegant method for
dealing with Gauss maps and dual hypersurface singularities using
Lagrange and Legendre geometry. The set L of oriented affine
lines in Euclidean space Rn is canonically
isomorphic with the symplectic manifold
T*Sn-1, the dual tangent
space of the unit (n-1)-sphere. An isomorphism from L
to the tangent space TSn-1 is
identified with T*Sn-1
using the Euclidean metric
(V -> <V, .>). An easy computation
shows that if
X: Mn-1 -> Rn
is an immersion, then the map N which assigns to PM the normal line to X at
P is a Lagrangian immersion of M in L. The Gauss
map is the Lagrangian map associated with the Lagrangian immersion
N, i.e. the composition of N with projection from
L = T*Sn-1
to Sn-1. (Cf. [A4, chapter 8 and appendix
12])

The contractification of the symplectic manifold
T*Sn-1 is the contact manifold
h consisting of all hyperplanes (contact elements) in the
tangent bundle of real projective space
RPn. (H can be identified with
PT*RPn, the
projectivized cotangent bundle of
RPn.) The Legendre fibration
N -> (RPn)* (the
dual projective space) assigns to each tangent hyperplane the
(n-1)-dimensional projective space to which it is tangent. If
X: Mn-1 -> RPn
is an immersion, then the map T which assigns to PM the tangent hyperplane to X
at P is a Legendre immersion of M in H. The dual
hypersurface
X*: M -> (RPn)*
is the Legendre map associated with T, i.e. the composition of
T with the Legendre fibration
N -> (RPn)*
(cf. [A4, appendix 4]).